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bootstrap_second_derivative_plus_previous.m
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bootstrap_second_derivative_plus_previous.m
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function [x, it] = bootstrap_second_derivative_plus_previous(inner_solver, target_alpha, v, R, tol, maxit, relspeed)
% Calls an inner solver iteratively over increasing values of alpha
% Predicts the new x using a 2nd order Taylor expansion plus the old x.
if not(exist('tol','var')) || isempty(eps)
tol = sqrt(eps);
end
if not(exist('maxit','var')) || isempty(maxit)
maxit = 10000;
end
if not(exist('relative_speed', 'var')) || isempty(relative_speed)
relative_speed = 0.01;
end
n = length(v);
total_iterations = 0;
alpha = nan;
old_alpha = nan;
new_alpha = 0.6;
x = nan(n, 1);
old_x = nan(n, 1);
while true
if any(isnan(old_x))
x_guess = v;
else
partialx = alpha*R*kron(eye(n),x) + alpha*R*kron(x,eye(n)) - eye(n);
partialalpha = R*kron(x,x) - v;
% from the implicit function theorem
xprime = -partialx \ partialalpha;
% now we differentiate (in alpha, total derivative) this expression for xprime piece by piece.
partialalpha_prime = R*kron(x, xprime) + R*kron(xprime, x);
partialx_prime = R*kron(eye(n),x) + R*kron(x,eye(n)) + alpha*R*kron(eye(n), xprime) + alpha*R*kron(xprime, eye(n));
% derivative of inv(partialx) =
% -inv(partialx)*partialx_prime*inv(partialx)
xsecond = -partialx \ (partialx_prime * xprime) - partialx \ partialalpha_prime;
h1 = new_alpha - alpha;
h2 = alpha - old_alpha;
x_guess = -(h1/h2)^3*old_x + x*(1+(h1/h2)^3) + xprime*(h1-h1^3/h2^2) + xsecond/2*(h1^2+h1^3/h2);
end
[alpha, old_alpha] = deal(new_alpha, alpha);
old_x = x;
[x, it] = inner_solver(alpha, v, R, tol, maxit-total_iterations, x_guess);
total_iterations = total_iterations + it;
if alpha >= target_alpha
break
end
% construct new alpha
if any(isnan(old_x))
% at the first step, we have no "second derivative" information
% available
new_alpha = alpha + 0.01;
else
second_derivative_guess = norm(x_guess - x) * 2 / norm(alpha - old_alpha)^2;
step_size = sqrt(2*relative_speed / second_derivative_guess);
new_alpha = alpha + step_size;
end
if new_alpha > target_alpha
new_alpha = target_alpha;
end
if total_iterations >= maxit
break
end
end
it = total_iterations;